### Unit 5: Rotational Kinematics

Now that we have discussed forces and how they manipulate motion, we will begin exploring a particular force that makes objects move in a curved motion. In this unit, we study the simplest form of curved motion: uniform circular motion, or motion in a circular path at a constant speed. In some ways, this unit is a continuation of the previous unit on dynamics, but we will introduce new concepts such as angular velocity and acceleration, centripetal force, and the force of gravity.

**Completing this unit should take you approximately 3 hours.**

Upon successful completion of this unit, you will be able to:

- explain why an object moving at a constant speed in a circle is accelerating;
- solve problems involving planets and satellites;
- explain what it means when an astronaut orbiting the Earth is described as being weightless;
- compare and contrast the physical properties associated with linear motion and rotational motion;
- state and explain Newton's Law of Universal Gravitation; and
- use kinematic equations to solve circular motion problems.

### 5.1: Centripetal Force

A centripetal force is any force that makes an object move in a circular motion. A centripetal force can involve any of the classical or electromagnetic forces, such as gravitational force (weight), normal force, tension, and friction.

For example, gravitational force acts as a centripetal force on a planetary scale because it causes planets to orbit in a circle. However, gravitational force is not a centripetal force on the Earth's surface because gravity makes objects fall straight down toward the Earth's center, not in a circle. A normal force can act as a centripetal force, such when a roller coaster does a loop-da-loop. Friction can act as a centripetal force, such as when it causes cars to turn corners on a road.

The equation for centripetal acceleration is , where is object speed and is radius (distance from center).

Read this text, which presents more explanation on the topic of centripetal acceleration. Pay attention to Figure 6.8 which shows an example of centripetal acceleration. In this example, a disk is rotating at a constant speed. As the disk rotates, the velocity vector at any given point on the disk changes because the direction changes. As shown in the free-body diagram at the top of the figure, the velocity vectors add to make a net velocity vector toward the center of the disk. This leads to centripetal acceleration because there is a net change in acceleration toward the disk.

Centripetal forces assume the equation . Therefore, whatever equation characterized the classical forces in Unit 4 can also be related to a situation by its centripetal force equation just given.

This lecture accompanies what you just read.

- Read this text, which demonstrates using the centripetal force equation in conjunction with the classical forces we learned in the previous unit: weight, normal force, tension, and friction.
- Watch this video for a demonstration of centripetal force.
This lecture accompanies what you just read.

Watch this video to see how the equation for centripetal force is derived.

Watch this video as it goes into the concept of centripetal acceleration and centripetal force.

Watch these two videos for examples of how to use normal force and gravity as a centripetal force in a loop-da-loop problem.

### 5.2: Centripetal Force and the Universal Law of Gravitation

A dramatic application of centripetal force is the Universal Law of Gravitation. Johannes Kepler (1571–1630), the German astronomer and mathematician, created three laws that pertain to orbital motion during the Renaissance period. At this time, physics and astronomy were two separate fields of study. Kepler developed these three laws independently from the laws of physics we know today.

Read this text, which includes visual diagrams of Kepler's Laws of Planetary Motion, which describe the motion of planets around the sun. We can also apply these laws to explain the motion of satellites around planets.

- Kepler's First Law of Planetary Motion states that planets move around the sun in an ellipse shaped orbit with the sun at the center of the ellipse (see Figure 6.29).
- Kepler's Second Law of Planetary Motion states that planets move so that a point on the planet sweeps an equal area in equal times (see Figure 6.30).
- Kepler's Third Law of Planetary Motion refers to the relationship between the time it takes for two planets to revolve around the sun, and their distances from the sun: , where and are periods of orbit while and are radii for planets one and two.

We can use Kepler's Third Law to solve problems to determine the period for planetary or satellite orbits. See a worked example of using the equation from Kepler's Third Law to determine the period of a satellite in Example 6.7. Pay attention to the derivation of Kepler's Third Law using the concept of centripetal forces.

- This video accompanies the text on Kepler's Three Laws.

### 5.3: Angular Position, Velocity, and Acceleration

In rotational motion, we deal with two-dimensional motion. Unlike with linear motion, we need to define angles and distances associated with circular motion.

Read this text. To understand circular or rotational motion, picture a spinning disk, such as the picture of a CD in Figure 6.2. This figure shows a CD with a line drawn from the center to the edge. All of the points along this line travel the same angle, in the same amount of time, as the CD spins. We call this the rotational angle, which is defined as . We call the distance along the circumference traveled () the arc length, and we call the radius of the circular motion () the radius of curvature.

When describing angles, we often use the unit radian, abbreviated as rad. We define radians as 1 revolution = rad. Radians are the standard unit for physics problems, but we can convert radians to the more familiar degrees for convenience. Pay attention to Table 6.1 for conversions between radians and degrees.

We define angular velocity (or rotational velocity), (the Greek letter omega), as the rate at which the angle changes while an object is rotating. We can write it as , where is the change in angle and is the time it takes for the angle to change that amount.

We can relate angular velocity to linear velocity using the relation , with being the radius of curvature. Pay attention to the derivation of how angular velocity relates to linear velocity in equations 6.6, 6.7, 6.8, and 6.9.

We define angular acceleration as the change in angular velocity with respect to time. The equation is , where represents angular acceleration.

As you read, pay attention to Example 10.1, which shows how to calculate the angular acceleration of a bike wheel. In the first part of the problem, we calculate the angular acceleration of the wheel given the change in angular velocity and time. In the second part of the problem, we calculate the time needed to stop an already spinning wheel given angular deceleration as initial velocity, using the same angular acceleration equation. See a diagram of a rotating object showing the relationship between linear and angular velocity in Figure 10.3.

### 5.4: Kinematics of Rotational Motion

Rotational Kinematics is the study of rotational motion, much like linear kinematics (or just plain kinematics) is the study of linear motion. When solving kinematics problems of rotational motion, we look at the relationships between angular and linear versions of position, velocity, and acceleration.

Read this text to see additional worked examples of how to solve problems involving the kinematics of rotational motion.

Example 10.3 shows how to calculate the kinematics of an accelerating fishing reel. Here, equation 10.19 is used to determine how the angular velocity changes with time. This result is used to calculate linear speed. Example 10.4 is an example where the fishing reel decelerates. Using equation 10.19, we solve for time rather than angular velocity. To see more of these types of problems, review Examples 10.5 and 10.6.

### Unit 5 Assessment

- Receive a grade
Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment